Relative Max ( 1st derivative test)
f'(x) = 0sign chart for f'(x)changes signPositive to negative
The first derivative test is a method that is used to determine the relative maximum and minimum points of a function. It involves taking the derivative of the function and analyzing its sign to identify where the function is increasing or decreasing. The points where the sign changes are known as critical points, which may be either local maxima or minima.
To identify a relative maximum using the first derivative test, you need to follow these steps:
1. Calculate the first derivative of the function.
2. Find the critical points by setting the first derivative equal to zero and solving for x.
3. Determine the sign of the derivative for each interval between the critical points by evaluating a point within each interval.
4. If the sign of the derivative changes from positive to negative at a critical point, then the point is a relative maximum. If the sign changes from negative to positive, then it is a relative minimum.
For example, let’s consider the function f(x) = x^3 – 3x^2 + 2x.
1. The first derivative of the function is f'(x) = 3x^2 – 6x + 2.
2. To find the critical points, we set f'(x) = 0 and solve for x.
3x^2 – 6x + 2 = 0
Using the quadratic formula, we get two solutions x = 0.422 and x = 1.578.
3. We choose test points within each interval to determine the sign of the derivative. For example, if we choose x = 0.5, we have f'(0.5) = 0.5 > 0, so the function is increasing on the interval (-∞, 0.422). If we choose x = 1, we have f'(1) = -1 < 0, so the function is decreasing on the interval (0.422, 1.578). If we choose x = 2, we have f'(2) = 10 > 0, so the function is increasing on the interval (1.578, ∞).
4. Since the sign of the derivative changes from positive to negative at x = 1.578, this point is a relative maximum.
Therefore, the relative maximum of the function f(x) = x^3 – 3x^2 + 2x is at (1.578, 1.381).
More Answers:
Discovering Relative Max Via Second Derivative TestMaster Local Extrema And Inflection Points With The Second Derivative Test: A Guide For Math Enthusiasts
Mastering The First Derivative Test: Finding Relative Maximums And Minimums Of Functions